CN-SMO 2015

Suppose that the sequence $\{a_n\}$ satisfy $a_1=1$ and $a_{2k}=a_{2k-1}+a_k, a_{2k+1}=a_{2k}$ for $k=1,2, \ldots$ Prove that $a_{2^n}< 2^{\frac{n^2}{2}}$ for any integer $n \geq 3$ .

Can you make $2015$ positive integers $1,2, \ldots , 2015$ to be a certain permutation which can be ordered in the circle such that the sum of any two adjacent numbers is a multiple of $4$ or a multiple of $7$ ?

In $\triangle ABC$ , we have $AB>AC>BC$ . $D,E,F$ are the tangent points of the inscribed circle of $\triangle ABC$ with the line segments $AB,BC,AC$ respectively. The points $L,M,N$ are the midpoints of the line segments $DE,EF,FD$ . The straight line $NL$ intersects with ray $AB$ at $P$ , straight line $LM$ intersects ray $BC$ at $Q$ and the straight line $NM$ intersects ray $AC$ at $R$ . Prove that $PA \cdot QB \cdot RC = PD \cdot QE \cdot RF$ .